Continuum Model for Coupled Acousto-Optical Phonons in Piezoelectric Materials

Abstract

A 3D continuum model is used to find optical and acoustic phonon fields in zincblende GaAs. We start out using continuum elastic differential equations and the Maxwell-Poisson equation to describe dynamic lattice strain and internal strain effects accounting for the full crystal symmetry of zincblende GaAs. The analytical model is derived in detail in a first-principles analysis. Our results reveal that for a slab of crystal GaAs grown along the [001] direction the mechanical displacements along the x and z directions $u_x, u_z$ couple while the mechanical displacement $u_y$ couple solely to the electric field E by virtue of piezoelectricity. As a consequence optical and acoustic phonon fields are inherently coupled due to piezoelectricity and acoustic and optical phonon modes must be found by solving simultaneously the full elastic solid and electric governing equations and the relevant (elastic and electric) boundary conditions. We then derive phonon dispersion curves for a GaAs slab and compare cases with and without anisotropy and piezoelectricity and show that neglecting the latter in the description of both acoustic and optical modes of GaAs, as is done in many classical descriptions, is a too crude approximation. We finally discuss two novel results: (i) confined coupled acousto-optical $u_y-u_\phi$ modes cannot exist in piezoelectric media except at certain discrete $q_x$ wavenumber values, and (ii) piezoelectricity prohibits the existence of optical phonon fields at the LO phonon frequency. The model presented is general and can be applied to other materials and other crystal structures.